BCFT: from the boundary to the bulk

The study of boundary conditions in rational conformal field theories is not only physically important. It also reveals a lot on the structure of the theory in the bulk''. The same graphs classify both the torus and the cylinder partition functions and provide data on their hidden quantum symmetry’'. The Ocneanu triangular cells – the 3j-symbols of these symmetries, admit various interpretations and make a link between different problems.